By R. M. R. Lewis

This e-book treats graph colouring as an algorithmic challenge, with a powerful emphasis on sensible purposes. the writer describes and analyses the various best-known algorithms for colouring arbitrary graphs, concentrating on even if those heuristics offers optimum suggestions sometimes; how they practice on graphs the place the chromatic quantity is unknown; and whether or not they can produce higher options than different algorithms for particular types of graphs, and why.

The introductory chapters clarify graph colouring, and boundaries and optimistic algorithms. the writer then indicates how complicated, glossy options should be utilized to vintage real-world operational study difficulties corresponding to seating plans, activities scheduling, and collage timetabling. He comprises many examples, feedback for extra examining, and old notes, and the ebook is supplemented by way of an internet site with an internet suite of downloadable code.

The booklet may be of price to researchers, graduate scholars, and practitioners within the parts of operations examine, theoretical desktop technology, optimization, and computational intelligence. The reader must have uncomplicated wisdom of units, matrices, and enumerative combinatorics.

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**Example text**

As a result, DS ATUR’s performance is more predictable. Indeed DS ATUR turns out to be exact for a number of elementary graph topologies. The ﬁrst of these is the bipartite graph, and to prove this claim it is ﬁrst necessary to show a classical result on the structure of these graphs. 8 A graph is bipartite if and only if it contains no odd cycles. Proof. Let G be a connected bipartite graph with vertex sets V1 and V2 . ) Let v1 , v2 , . . , vl , v1 be a cycle in G. We can also assume that v1 ∈ V1 , v2 ∈ V2 , v3 ∈ V1 , and so on.

Consider further a permutation π of G’s vertices that has been generated such that the vertices occurring in each colour class of S are placed into adjacent locations in π. If we now use this permutation with G REEDY, the result will be a new solution S that uses no more colours than S, but possibly fewer. 1 The Greedy Algorithm (a) 31 v1 v2 v3 v5 (b) v1 v2 v4 v3 v4 v6 v5 v6 v7 v8 v7 v8 v9 v10 v9 v10 Fig. 1 Let S be a feasible colouring of a graph G. If each colour class Si ∈ S (for 1 ≤ i ≤ |S|) is considered in turn, and all vertices are fed one by one into the greedy algorithm, the resultant solution S will also be feasible, with |S | ≤ |S|.

In this chapter we will consider three fast constructive methods which operate by assigning each vertex to a colour one at a time using rules that are intended to keep the overall number of colours as small as possible. As we will see, for certain graph topologies some of these algorithms turn out to be exact, though in most cases they only produce approximate solutions. The ﬁrst of these algorithms, the so-called G REEDY algorithm, is perhaps the most fundamental method in the ﬁeld of graph colouring and is also useful for establishing bounds on the chromatic number.